I have been reading The Polynomial Method for Random Matrices by N. Raj. Rao and A. Eldeman as a part of literature review for my PhD research. In this paper, the authors present a method to determine the limiting eigen distribution of a particular subset of random matrices called algebraic random matrices. In the proposed method, the Stieltjes transform of the limiting eigen distribution is encoded as a root of a bivariate polynomial. A defined set of transformations ( deterministic and stochastic ) on the matrix is mapped to an operation on the bivariate polynomial. The limiting eigen distribution of the resulting matrix can be determined from the new bivariate polynomial.
This method heavily relies on the symbolic computation methodology. In parallel with the development of the mathematical framework for the polynomial method, the authors have developed a Matlab toolbox called the RMTool which leverages the symbolic computation capability of Matlab. However the RMTool uses the Maple Symbolic Toolbox for Matlab which Mathworks no longer supports (Above version 2009a). The original RMTool might even have been written in Maple itself and later forked to a Matlab toolbox. This has made RMTool extremely platform dependent tool preventing it from being used by other researchers. The method was proposed sometime in 2005 but I can hardly find an alternative reference on this topic or its application. It seems me that the rigidity of the RMTool may be limiting its potential as a powerful tool in random matrix analysis and application to engineering problems.
I have been looking at possibilities of porting the RMTool to the new Matlab Symbolic Toolbox which is based on MUPad. I am also planning on developing a new package ( from scratch :o) on a more robust and compatible platform ( Python with Sage ). But these ideas are still in early phase and need a thorough understanding of the polynomial. method.
Python implementation for finding Nth prime number#!/usr/bin/python # thousandPrime.py : Finds the 1000th prime number, fork of Problem Set 1 MIT 6.00 # Programmed: Saurav R Tuladhar # Date: Oct 7, 2011 # Declare state variables counter = 1; # Counts number of primes idx = 1; testNum = 2*idx + 1; # Odd numbers > 2 as candidates primeList = ; isPrime = True while counter < 10000: for x in primeList: if testNum % x == 0: # Only check for prime numbers < testNum. Based on Fundamental Theorem of Arithmetic. isPrime = False break if isPrime == True: primeList = primeList + [testNum] counter = counter + 1 # Reset variables isPrime = True idx = idx + 1 testNum = 2*idx + 1 print primeList[-1]
I am just back from the Underwater Acoustic Signal Processing Workshop 2011 held at W. Alton Jones Campus, University of Rhode Island. It was a three day conference from 12th – 14th October, 2011.
The chairman of the conference was my advisor Dr. John R. Buck. My lab partner David Hague gave a talk on his Compressed Sensing based active SONAR model inspired by bat’s biosonar capability.
Here are my reflections on the conference:
- The conference began with a reception banquet dinner where G. Clifford Carter was awarded the UASP Award. Apparently it turns out that G. C. Carter invented the Generalized Cross Correlation (GCC) method for time delay estimation. The acronym for this method matches the initials of Carter’s name.
- Although the conference was on underwater signal processing, there were three plenary sessions on underwater autonomy which mainly dealt with robotics and control systems oriented design problems for underwater deployment of autonomous vehicles. I was a bit disappointed to see very less of signal processing. However where were one session each on Array processing , Noise Modeling and Acoustics Communications which were in the ball park of my interest.
- The navy seems to have a huge interest in developing unmanned underwater autonomous vehicles and there a lots of companies and academic laboratories working on this area. I am not particularly interested on the navy’s perspective on this, but as far as I understand the systems development has largely shifted towards being software based design.
- Large fraction of presentations were focused on military (navy) applications or the signal processing problems they were trying to solve were from military applications point of view. The focus on military applications was a bit too much for my liking.
- Certainly there are some civilian applications of the results from these research.
- There were few presentations on Synthetic Aperture SONAR (SAS) and I came to an understanding that synthetic aperture is analogous to taking multiple photographs and stitching them together to form a panorama.
- There was a presentation by Aurther Baggeror on why MFP failed. My perception was that no body was sure why this particular method failed, but they already knew it had died.
- Interesting discussion on Coherence, brought up by Henry Cox.
- It was satisfying to see a large fraction of presentations using real field data for validation of their results. In computer simulations everything works :D.
The Fall 2011 semester officially commenced from 7th September. In the beginning of the summer I had already registered for a course on Discrete Mathematics ( MTH 550) and also planned to audit the CIS course on Algorithms and Data Structures.. I am also registered for Abstract Algebra (MTH 441) as suggested by my advisor. In addition to all that I am also attending the course on Random Signals and Systems being taught by my advisor.
Last Friday I finally submitted the revised paper to JASA. During Spring 2011, I and my adviser wrote a journal paper to publish the findings of my MS research on optimum sensor array design. The first manuscript of the paper was submitted to JASA in May 2011. The paper was partially accepted with few comments from the reviewer.
We were able to reply to all the comments but one. I had an intuitive feel to the solution but did not have a rigorous proof to back my intuition. My adviser always says we should follow our intuition and back it up with a rigorous proof. And over the past week I eventually came up with a mathematical proof and we were able to compose a solid response for the revised submission of the paper.
The paper has been published in JASA Vol 130, Issue 5. The permlink to the paper is http://dx.doi.org/10.1121/1.3644914